358 AUDREY TERRAS, ARITHMETICAL QUANTUM CHAOS
• •
H
T
Figure 1. The 3 Symmetric Spaces. The Poincare upper half plane H is on
the upper left. The 3-regular tree T is on the bottom, and the finite upper
half plane H3 over the field with 3 elements is on the upper right.group is the general linear group G = GL(2,1Fq) which consists of 2 x 2 matrices
with entries in the finite field and non-0 determinant.
For the second column, we will emphasize the graph theory rather than the
group theory as that would involve p-adic groups which would require more back-
ground of the reader and not even include the most general degree graphs. See Li
[ 49] or N agoshi [ 63] for the p-adic point of view. Thus for the ( q + 1 )-regular tree
we view the group Gas a group of graph automorphisms, meaning 1-1, onto maps
from T to T preserving adjacency. There are 3 types of such automorphisms of T.
See Figa-Talamanca and Nebbia [29] for a proof.
Automorphisms of the tree T
- rotations fix a vertex;
- inversions fix an edge and exchange endpoints;
- hyperbolic elements p fix a geodesic {xnln E Z} and p(xn) = Xn+s· That
is, p shifts along the geodesic bys = v(p). We define "geodesic" below.
One is pictured in Figure 3.
The subgroup K of g E G = SL(2, IR) such that gi = i is easily seen to be
the special orthogonal group 80(2, IR) of 2 x 2 rotation matrices. The analogue
for G = GL(2, lFq) consists of elements fixing the origin ,/8 in the finite upper half
plane Hq and it is the subgroup K of matrices of the form ka,b = ( ~ b: ). It
is easily seen that the map sending ka,b to a + b,/8 provides a group isomorphism
from K to the multiplicative group of lF q ( ,/8).